Limiting distributions of theta series on Siegel half-spaces

Authors:
F. Götze and M. Gordin

Translated by:

Original publication:
Algebra i Analiz, tom **15** (2003), nomer 1.

Journal:
St. Petersburg Math. J. **15** (2004), 81-102

MSC (2000):
Primary 11Fxx, 37D30

DOI:
https://doi.org/10.1090/S1061-0022-03-00803-3

Published electronically:
December 31, 2003

MathSciNet review:
1979719

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an integer. For any in the Siegel upper half-space we consider the multivariate theta series

The function is invariant with respect to every substitution , where is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix the function can be viewed as a complex-valued random variable on the torus with the Haar probability measure. It is proved that the weak limit of the distribution of as exists and does not depend on the choice of . This theorem is an extension of known results for to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of Dani-Margulis' and Ratner's results on dynamics of unipotent flows.

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Additional Information

**F. Götze**

Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Email:
goetze@mathematik.uni-bielefeld.de

**M. Gordin**

Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191011, Russia

Email:
gordin@pdmi.ras.ru

DOI:
https://doi.org/10.1090/S1061-0022-03-00803-3

Keywords:
Theta series,
Siegel's half-space,
convergence in distribution,
closed horospheres,
unipotent flows

Received by editor(s):
September 2, 2002

Published electronically:
December 31, 2003

Additional Notes:
Supported in part by the DFG-Forschergruppe FOR 399/1-1.

M. Gordin was also partially supported by RFBR (grant no. 02.01-00265) and by Sc. Schools grant no. 2258.2003.1. He was a guest of SFB-343 and the Department of Mathematics at the University of Bielefeld while the major part of this paper was prepared.

Article copyright:
© Copyright 2003
American Mathematical Society